Solov'ev's Solution

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Solov'ev's Solution is the simplest two dimensional analytical solution to the inhomogeneous Grad–Shafranov equation.

As a reminder, the so called source functions of the Grad-Shafranov equation are defined by F = rBT, with r the radial distance from the axis of symmetry, and p, the pressure. ψ represents the flux function.

In Solov'ev's solution the source functions are defined simply as

p(ψ) = − p'ψ + p0

and

F^2(\psi) = F_0^2 + \frac{2 \gamma p'}{1 + \alpha^2} \psi

Where a prime denotes derivation by ψ, and γ and α are constants. When γ is set to 0, there is no toroidal field, yielding a Field Reversed Configuration.

Solving the Grad-Shafranov equation, we get

\nabla^* \psi = p' \left({r^2 - \frac{\gamma}{1 - \alpha^2}}\right)

Yielding the solution

\psi = \frac{p'}{2 (1 - \alpha^2)} \left[{(r^2 - \gamma) Z^2 + \frac{\alpha^2}{4} (r^2 - r^2_0)^2 }\right]

Solov'ev's solution is simple and exact, but there is afforded little flexibility in the current distribution, making it difficult to fit to many systems of interest.

== Reference ==

S. Solov’ev, Sov. Phys. JETP 26, 400 (1968); Zh. Eksp. Teor. Fiz. 53, 626 (1967).

This page was recovered in October 2009 from the Plasmagicians page on Solov'ev's_Solution dated 21:00, 6 July 2007.

Retrieved from "https://qed.princeton.edu/main/PlasmaWiki/Solov%27ev%27s_Solution"
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